We present a theoretical description of surface plasmon polariton(SPP)on a graphene surface excitated by four-wave mixing using two laser beams.The phase-matched incident angles of the two beams that meet the SPP excitation conditions are calculated,and the discussion of the generated SPP power dependence on the incident angles is presented.We demonstrate that there is an enhancement in the peak power at the optimum angles at 633 nm and 921 nm,which is about 30 and 850 times larger than the corresponding peak power on a gold surface,respectively.
We study the electronic properties for the graphene-based one-dimensional superlattices, whose potential voltages vary according to the envelope of a Gaussian function. It is found that an unusual Dirac point exists and its location is exactly associated with a zero-averaged wave number （zero-re） gap. This zero-k gap is less sensitive to incident angle and lattice constants, properties opposing those of Bragg gap. The defect mode appearing inside the zero-l gap has an effect on transmission, conductance, and shot noise, which will be useful for further investigation.
Recently, a new （2＋1）-dimensional shallow water wave system, the （2＋1）-dlmenslonal displacement shallow water wave system （2DDSWWS）, was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS, which is the same as the famous Kadomtsev-Petviashvili equation and Korteweg-de Vries equation. Applying dimensional analysis, we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS （M2DDSWWS） is studied and four similarity solutions are obtained. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the M2DDSWWS will degenerate to the 2DDSWWS.
Novel band-stop filters with circular split-ring resonators based on the metal-insulator-metal （MIM） structure are presented, with their transmission properties of SPPs propagating through the filter simulated by the finite-difference time-domain （FDTD） method, The variation of the gap of the split ring can affect the transmission characteristics, i.e., the transmission spectrum of SPPs exhibiting a shift, which is useful for modulating the filter. Linear and nonlinear media are used in the resonator respectively. By varying the refractive index of the linear medium, the transmission properties can be changed obviously, and the effect caused by changing the incident intensity with a nonlinear medium is similar. Several resonant modes that are applicable can be enhanced by changing the position of the gap of the split ring. Thus, the transmission properties can be modulated by adjusting the size of the gap, varying the refractive index, and changing the incident intensity of the input light. These methods may play significant roles in applications of optical integrated circuits and nanostructural devices.
